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Chapter 16: Problem 3

If \(\gamma=\left(\begin{array}{llll}1 & 2 & 3 & 4 \\ 2 & 1 & 4 & 3\end{array}\right) \in S_{4}\), how many cosets does \(\langle\gamma\rangle\) determine?

Short Answer

Expert verified
The number of cosets determined by \( \langle\gamma\rangle \) in \( S_{4} \) is 12.

Step by step solution

01

Understanding the given permutation

First, understand the given permutation. The permutation \( \gamma=\left(\begin{array}{llll}1 & 2 & 3 & 4 \ 2 & 1 & 4 & 3\end{array}\right) \) can be written in cycle notation as \( \( \gamma = (1 2)(3 4) \) \), which is a product of 2 disjoint 2-cycles.
02

Computing the order of the permutation

The next step is to compute the order of the given permutation. Remember that the order of a permutation is the least common multiple of the lengths of its disjoint cycles. Since \( \gamma \) is composed of two 2-cycles, its order is 2.
03

Computing the number of cosets

The order of the permutation \( \gamma \) serves as the order of the subgroup \( \langle\gamma\rangle \). This subgroup divides \( S_{4} \) into cosets. As \( S_{4} \) has \( 4! = 24 \) elements and the order of \( \langle\gamma\rangle \) is 2, by the Lagrange's Theorem (the order of a subgroup divides the order of the group), there must be \( 24/2 = 12 \) cosets determined by \( \langle\gamma\rangle \) in \( S_{4} \)

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Most popular questions from this chapter

Let \(p\) be a prime. (a) If \(G\) has order \(2 p\), prove that every proper subgroup of \(G\) is cyclic. (b) If \(G\) has order \(p^{2}\), prove that \(G\) has a subgroup of order \(p\).

Let \(G\) be a group where \(a^{2}=e\) for all \(a \in G\). Prove that \(G\) is abelian.

Find all subgroups in each of the following groups. a) \(\left(\mathbf{Z}_{12},+\right)\) b) \(\left(\mathbf{Z}_{11}^{*}, \cdot\right)\) c) \(S_{3}\)

In \(S_{5}\) find an element of order \(n\), for all \(2 \leq n \leq 5\). Also determine the (cyclic) subgroup of \(S_{5}\) that each of these elements generates.

Let \(G\) be a group with subgroups \(H\) and \(K\). If \(|G|=660,|K|=66\), and \(K \subset H \subset G\), what are the possible values for \(|H|\) ?
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